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In Section 21 of "Quantum Field theory" by Mark Srednicki, it is shown that there are two equivalent ways to get the quantum action of the shifted field $\phi'= \phi-\tilde{\phi}$, where $\phi$ is the original field and $\tilde{\phi}$ is the background. (See Eq. (21.27)) One way is to first perform the shift at the classical level and then derive the quantum action of $\phi'$ by treating $\tilde{\phi}$ as a new parameter. The other is to first derive the quantum action of $\phi$ and then perform the background shift.

Similarly, the renormalization group (RG) equations of all the parameters in the shifted action $S'$ can be derived in those two ways. The first way is to directly derive the RG equations for $S'$. The second way is to first derive the RG equations for the original action, and then get the RG equations for $S'$ using the parameter relations given by the background shift. My question is: do these two ways lead to the same RG equations?

Please let me know if I'm not clear enough.

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